Theorem subgex 13708

Description: The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)

Assertion

Ref	Expression
subgex	|- ( G e. Grp -> (SubGrp ` G ) e. _V )

Proof of Theorem subgex

Dummy variables s w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-subg 13702	. . 3 |- SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
2		fveq2 5626	. . . . 5 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
3	2	pweqd 3654	. . . 4 |- ( w = G -> ~P ( Base ` w ) = ~P ( Base ` G ) )
4		oveq1 6007	. . . . 5 |- ( w = G -> ( w ↾s s ) = ( G ↾s s ) )
5	4	eleq1d 2298	. . . 4 |- ( w = G -> ( ( w ↾s s ) e. Grp <-> ( G ↾s s ) e. Grp ) )
6	3, 5	rabeqbidv 2794	. . 3 |- ( w = G -> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } = { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } )
7		id 19	. . 3 |- ( G e. Grp -> G e. Grp )
8		basfn 13086	. . . . . 6 |- Base Fn _V
9		elex 2811	. . . . . 6 |- ( G e. Grp -> G e. _V )
10		funfvex 5643	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5422	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
12	8, 9, 11	sylancr 414	. . . . 5 |- ( G e. Grp -> ( Base ` G ) e. _V )
13	12	pwexd 4264	. . . 4 |- ( G e. Grp -> ~P ( Base ` G ) e. _V )
14		rabexg 4226	. . . 4 |- ( ~P ( Base ` G ) e. _V -> { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } e. _V )
15	13, 14	syl 14	. . 3 |- ( G e. Grp -> { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } e. _V )
16	1, 6, 7, 15	fvmptd3 5727	. 2 |- ( G e. Grp -> (SubGrp ` G ) = { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } )
17	16, 15	eqeltrd 2306	1 |- ( G e. Grp -> (SubGrp ` G ) e. _V )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2799   ~Pcpw 3649   Fn wfn 5312   ` cfv 5317  (class class class)co 6000   Basecbs 13027   ↾_s cress 13028   Grpcgrp 13528  SubGrpcsubg 13699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-ov 6003  df-inn 9107  df-ndx 13030  df-slot 13031  df-base 13033  df-subg 13702

This theorem is referenced by:  isnsg  13734